"The chain's home base is the University of Toledo Medical Center, where transplant surgeon Mike Rees performs operations and coordinates the program through his nonprofit, the Alliance for Paired Donation. People enter the chain because they need a kidney and have a friend or family member who is willing to donate, but who isn't a match. Once they enter the chain, Rees's staff inputs their names, blood types and other information into a database of other patient-and-donor pairs; the computer then matches would-be donors and recipients."
The article is referring to a recent reunion of the 20 people involved in Rees' first pioneering non-simultaneous extended altruistic donor (NEAD) chain.
Another organization that has been successfully pursuing NEAD chains among other options is the National Kidney Registry founded by Garet Hil. Here's an article about a recent exchange of theirs: Two couples from Bronx, Jersey exchange kidneys through computer organ donation program.
See here for some more technical material on kidney exchange.